Magnetic resonance imaging is based on the process of inverting the spins of atoms which are situated in a strong axial magnetic field and then measuring the electromagnetic radiation of the atoms, as the spins return to a more relaxed state. A practical MRI device requires the ability to selectively invert a narrow slice of a subject, in a short period of time and using a low dose of RF radiation. The usual manner of inversion includes applying a z-gradient magnetic field to the subject so that each slice of the subject has a different Larmor frequency and irradiating the subject with an RF radiation pulse, so that only the spins in one of these slices are inverted. As higher strength magnetic fields are used for MRI imaging, the amount of RF energy absorbed by the body is higher. It is therefore important to limit the amount of radiation to which the subject is exposed. Further, in many MRI devices, the peak RF amplitude is limited. Usually, there is a tradeoff made between the pulse duration and the RF amplitude.
The relationship between the RF radiation, the magnetic field and the inversion of the spins is described by the Bloch equations, for which there are only a small number of known analytical solutions.
When an RF electromagnetic field is applied to a spin which is already in a strong static magnetic field, the RF magnetic field affects the spin. The RF field is very much smaller than the static field, so the RF field is usually described as rotating in the plane perpendicular to the field direction of the static magnetic field (the effect of the component in the static field direction is negligible). The effect of the RF field on the spins is most conveniently described in a rotating frame of reference, having three perpendicular axes, Z, Y and X. The Z axis is aligned with the main magnetic field denoted by M.sub.z. The X axis is aligned with the RF field and the Y axis is perpendicular to both the X and Z axes. The entire frame of reference rotates around the Z axis at the instantaneous angular (frequency) of the RF pulse. Both X and Z axes use units of frequency, such that all magnetic fields B are represented by vectors .gamma.B, where .gamma. is the gyro-magnetic resonance coefficient for the spin (type of species thereof).
The effective magnetic field to which a spin is subjected as a result of the RF field is preferably defined as a vector in the rotating frame of reference. The magnitude of the Z component of the vector is equal to the frequency difference between the RF field frequency and the Larmor frequency of the spin. The magnitude of the X component is equal to the instantaneous amplitude of the RF field. It should be appreciated that in a uniform Z directed field, all the spins are located at the same Z coordinate. When a gradient magnetic field is applied, each spin has a different Larmor frequency and, hence, a different Z coordinate.
Typically, the net magnetization of a group of spins is treated as a single vector value, called the magnetization vector. Thus, the effect of an inversion pulse is to invert the magnetization vector in a slice of tissue. FIG. 1 is a graph of a typical inverted slice profile in which a nomalized magnetization is shown as a function of an off-resonance frequency. The slice includes an in-slice region, which is inverted by the inversion pulse, an out-of-slice region which is not inverted by the pulse and a transition region where the post-inversion magnetization varies between +1 (not inverted) and -1 (inverted). The magnetization values are normalized to the equilibrium magnetization, M.sub.0. For convenience, the in-slice region is usually depicted as centered around the magnetization axis. The width of the slice (SW) is usually measured between the two points where the post-inversion magnetization vector is zero. The slice width is measured in units of frequency, which reflect the relationship between the inversion and the Larmor frequency. For convenience, C.sub.o is defined to be half the transition width.
One important type of inversion pulse is an adiabatic pulse. Inversion by adiabatic pulse is less affected by inhomogenities of the RF field amplitude than is inversion by other types of inversion pulses. An adiabatic pulse uses the following mechanism: An effective magnetization of the RF radiation field is initially aligned with the main field magnetization axis (+M.sub.z) direction and is slowly changed until it is aligned in the direction opposite the main field magnetization (-M.sub.z). If the rate of change of the effective magnetization vector is gradual enough, the magnetization vector will track the effective magnetization of the RF field and will be inverted when the effective magnetization vector becomes aligned with the -Z axis. The adiabatic condition (described below) describes the conditions under which the rate of change of the vector is sufficiently gradual to permit tracking. The motion of the effective magnetization is characterized by its "trajectory", which is the path of the tip of the effective magnetization vector and its "velocity profile", which describes the instantaneous rate of motion of the effective magnetization vector along its trajectory.
FIG. 2 is a graph showing the trajectory of a typical adiabatic pulse in the Z-X plane. The effective magnetization vector of the pulse starts out aligned with the +Z direction and moves along a half ellipse in the Z-X plane until it becomes aligned with the -Z direction. It should be noted that the trajectory shown in FIG. 2 is only correct for spins at the center of the slice. For all other spins, the shown trajectory is shifted by an amount equal to the difference between the Larmor frequency of the spin and the Lartnor frequency at the slice center. For each point P along the trajectory, which indicates an instantaneous position of the effective magnetization vector, x is the instantaneous RF amplitude and z is the instantaneous RF synthesizer frequency. For each spin which is affected by the adiabatic pulse, a vector connecting the spin and point P is the effective field vector, having a magnitude r. .theta. is defined as the angle between r and the X axis. In order for the rate of change of the vector to be sufficiently gradual to permit tracking, the motion must satisfy the following (adiabatic) condition, .GAMMA.=r/.vertline..theta..vertline.&gt;&gt;1, where .GAMMA. is an adiabatic parameter which describes this ratio. For the same magnetization vector traversing a given trajectory at a given rate of motion, different spins will see different angular velocities. Since r and .theta. are different for each spin, the adiabatic parameter may ensure tracking for one group of spins but not for another, even at the same point P.
As can be appreciated, if .theta. is larger, the pulse will be shorter, however, the adiabatic parameter will be smaller, so tracking may break down and not be possible. In some MRI imaging sequences, time is of essence, so a short inversion pulse is desired.
One of the most efficient (fast, low peak RF amplitude and adiabatic for a wide range of RF amplitudes) inversion pluses in the prior art is the sech/tanh pulse. The first term (sech) defines the X component of the magnetization vector and the second term (tanh) describes the Z component. The trajectory of the sech/tanh pulse is a half ellipse in the Z-X plane.
"General Solutions for Tailored Modulation Profiles in Adiabatic Excitation", by Thomas E. Skinner and Pierre-Marie L. Robitaille, published in the Journal of Magnetic Resonance 98, pp. 14-23 (1992), describes an inversion pulse having a triangular trajectory. FIG. 3 shows an example of such a trajectory.
"Single-Shot, B1-Insensitive Slice Selection with a Gradient-Modulated Adiabatic Pulse, BISS-8", by Robin A. de Graaf, Klaas Nicolay and Michael Garwood, published in Magnetic Resonance in Medicine 35:652-657 (1996), describes a method for generating an optimal slice-selection pulse, named BISS-8, having an adjustable flip angle. A main benefit of the BISS-8 pulse is that it does not scramble the phase of the selected slice (which most adiabatic pulses do), so it can also be used for 180 refocusing in spin-echo imaging. The BISS-8 pulse requires much more amplitude than comparable pulses. However, the peak required amplitude is lower than comparable pulses. In addition, both the gradients and the RF frequency are modified during a BISS-8 pulse.